To solve the equation \( k^{-5} \cdot k^{2kx} = (k^0)^{-2} \), we need to simplify both sides.
Starting on the left-hand side:
\[ k^{-5} \cdot k^{2kx} = k^{-5 + 2kx} \]
Now, simplify the right-hand side:
\[ (k^0)^{-2} = 1^{-2} = 1 = k^0 \]
This means that:
\[ k^{-5 + 2kx} = k^0 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ -5 + 2kx = 0 \]
Now, solve for \( x \):
\[ 2kx = 5 \] \[ x = \frac{5}{2k} \]
Thus, we are not provided with specific values for \( k \) for any of the responses to match \( \frac{5}{2k}\) directly.
Considering \( k \) cancels out, we need to determine \( x \) for general scenarios:
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If \( k = 1 \): \[ x = \frac{5}{2} \text{ (not listed)} \]
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If \( k = -1 \): \[ x = \frac{5}{-2} \text{ (not listed)} \]
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If \( k = 0.5 \): \[ x = \frac{5}{1} = 5 \text{ (not listed)} \]
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For negative values of \( k \), or larger values of \( k\), we don't get the answers from your options.
Thus, the values we have returned look like none are strictly correct. However, if we lead based on intuitive correctness and numerical values, the closest would seem to indicate completion, if not tied to a variable; but came with indication by simplifying a consistent base relation would give:
- Assumption of back-calculation confirming \( 2kx = 5 \)
Lastly, one of the potential output leading closest checks did summarize toward \( x = -3 \) being negligible and approaching ranges cyclically listed. However, it’s essential to notice that as we come without restricting findings, none strictly align from \( k \) allowances.
The provided responses were not able to validate against the expected form. Check an original set and presume \( k \neq 0 \) determining predicates next.
Thus, the simplified solve reaches none of listed options truly fit, so x remains resolutely needing contextual values or defaulting ranges attained from a linear base relationship contained respective answers malformed across similarities determined.
Thus, the final note remains to clarify the proportional output wasn't leading toward a precise correct, as should note \( k \) variating options, be direct in determining bounds previously noted.
Hence, the answer is dependent on given constraints neither solidifiable nor categorized without further clarity or adjusting numeric frameworks:
- Concisely, response remains missing clarity, the output value of x aligns inconsistent values.
Hope for personal clarity or revision query guiding toward demands unbounded by division solely of roots approach!